Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity:
for complex numbers x and y, with |x| < 1 and y ≠ 0.
The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.
Let and . Then we have
The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:
Then the Jacobi theta function
can be written in the form
Using the Jacobi Triple Product Identity we can then write the theta function as the product
There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols:
where is the infinite q-Pochhammer symbol.
It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For it can be written as
This proof uses a simplified model of the Dirac sea and follows the proof in Cameron (13.3) which is attributed to Richard Borcherds. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as
A level is a half-integer. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The energy of the state is
and the particle number of is
An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function for the number of states of energy with particles can be expressed as
On the other hand, any state with particles can be obtained from the lowest energy particle state, , by rearranging particles: take a partition of and move the top particle up by levels, the next highest particle up by levels, etc.... The resulting state has energy , so the generating function can also be written as
- See chapter 14, theorem 14.6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
- Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0
- Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012